Properties of Hermitian Matrices

Section 5.3 Properties of Hermitian Matrices

The eigenvalues and eigenvectors of Hermitian matrices Section 5.2 have some special properties.

The eigenvalues of a Hermitian matrix must be real.

To see why this relationship holds, start with the eigenvector equation

\begin{equation} M |v\rangle = \lambda |v\rangle\tag{5.3.1} \end{equation}

and multiply on the left by \(\langle v|\) (that is, by \(v^\dagger\)):

\begin{equation} \langle v | M | v \rangle = \langle v | \lambda | v \rangle = \lambda \langle v | v\rangle\text{.}\tag{5.3.2} \end{equation}

But we can also compute the Hermitian conjugate (that is, the conjugate transpose) of (5.3.1), which is

\begin{equation} \langle v| M^\dagger = \langle v| \lambda^*\text{.}\tag{5.3.3} \end{equation}

Using the fact that \(M^\dagger=M\text{,}\) and multiplying by \(|v\rangle\) on the right now yields

\begin{equation} \langle v | M | v \rangle = \langle v | \lambda^* | v \rangle = \lambda^* \langle v | v \rangle\text{.}\tag{5.3.4} \end{equation}

Comparing (5.3.2) with (5.3.4) now shows that

\begin{equation} \lambda^* = \lambda\tag{5.3.5} \end{equation}

as claimed.

Eigenvectors corresponding to different eigenvalues must be orthogonal.

The argument establishing this relationship is similar to the one above. Suppose that

\begin{align} M |v\rangle \amp = \lambda |v\rangle ,\tag{5.3.6}\\ M |w\rangle \amp = \mu |w\rangle\text{.}\tag{5.3.7} \end{align}

Then

\begin{equation} \langle v | \lambda | w \rangle = \langle v | M | w \rangle = \langle v | \mu | w \rangle\tag{5.3.8} \end{equation}

or equivalently

\begin{equation} (\lambda - \mu) \langle v | w \rangle = 0\text{.}\tag{5.3.9} \end{equation}

Thus, if \(\lambda\ne\mu\text{,}\) \(|v\rangle\) must be orthogonal to \(|w\rangle\text{.}\)

Repeated Eigenvalues.

In the case of repeated eigenvalues, it is possible to make a basis orthogonal using a process called Gram-Schmidt orthogonalization. (This amounts to simply subtracting off the parts of a given basis that are not orthogonal.)

To Remember.

It is always possible to find an orthonormal basis of eigenvectors for any Hermitian matrix.

Sensemaking 5.1. Properties of Anti-Hermitian Matrices.

Repeat the two proofs above to find similar properties for anti-Hermitian matrices.

Hint 1.

Don’t forget to use \(M^{\dagger}=-M\text{.}\)

Hint 2.

Don’t forget to complex conjugate the eigenvalue, where necessary.

Answer.

The eigenvalues are pure imaginary. The eigenvectors corresponding to different eigenvalues are orthogonal.

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